Definition:Totally Bounded Metric Space/Definition 1
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Definition
A metric space $M = \struct {A, d}$ is totally bounded if and only if:
- for every $\epsilon \in \R_{>0}$ there exists a finite $\epsilon$-net for $M$.
That is, $M$ is totally bounded if and only if:
- for every $\epsilon \in \R_{>0}$ there exists a finite set of points $x_1, \ldots, x_n \in A$ such that:
- $\ds A = \bigcup_{i \mathop = 1}^n \map {B_\epsilon} {x_i}$
- where $\map {B_\epsilon} {x_i}$ denotes the open $\epsilon$-ball of $x_i$.
That is: $M$ is totally bounded if and only if, given any $\epsilon \in \R_{>0}$, one can find a finite number of open $\epsilon$-balls which cover $A$.
Also known as
A totally bounded metric space is also referred to as a precompact space.
Also see
- Results about totally bounded metric spaces can be found here.
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $7.2$: Sequential compactness: Definition $7.2.10$
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): totally bounded